Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 021, 41 pages      arXiv:2108.02608      https://doi.org/10.3842/SIGMA.2023.021

Rank 4 Nichols Algebras of Pale Braidings

Nicolás Andruskiewitsch a, Iván Angiono a and Matías Moya Giusti b
a) Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, CIEM - CONICET, Medina Allende s/n (5000) Ciudad Universitaria, Córdoba, Argentina
b) 6 rue Rampal, 75019, Paris, France

Received November 25, 2022, in final form March 21, 2023; Published online April 13, 2023

Abstract
We classify finite GK-dimensional Nichols algebras ${\mathscr B}(V)$ of rank 4 such that $V$ arises as a Yetter-Drinfeld module over an abelian group but it is not a direct sum of points and blocks.

Key words: Hopf algebras; Nichols algebras; Gelfand-Kirillov dimension.

pdf (725 kb)   tex (45 kb)  

References

  1. Andruskiewitsch N., An introduction to Nichols algebras, in Quantization, Geometry and Noncommutative Structures in Mathematics and Physics, Math. Phys. Stud., Springer, Cham, 2017, 135-195.
  2. Andruskiewitsch N., On pointed Hopf algebras over nilpotent groups, Israel J. Math., to appear, arXiv:2104.04789.
  3. Andruskiewitsch N., Angiono I., On Nichols algebras with generic braiding, in Modules and Comodules, Trends Math., Birkhäuser, Basel, 2008, 47-64, arXiv:math.QA/0703924.
  4. Andruskiewitsch N., Angiono I., On finite dimensional Nichols algebras of diagonal type, Bull. Math. Sci. 7 (2017), 353-573, arXiv:1707.08387.
  5. Andruskiewitsch N., Angiono I., Heckenberger I., On finite GK-dimensional Nichols algebras of diagonal type, in Tensor Categories and Hopf Algebras, Contemp. Math., Vol. 728, Amer. Math. Soc., Providence, RI, 2019, 1-23, arXiv:1803.08804.
  6. Andruskiewitsch N., Angiono I., Heckenberger I., On finite GK-dimensional Nichols algebras over abelian groups, Mem. Amer. Math. Soc. 271 (2021), ix+125 pages, arXiv:1606.02521.
  7. Andruskiewitsch N., Heckenberger I., Schneider H.J., The Nichols algebra of a semisimple Yetter-Drinfeld module, Amer. J. Math. 132 (2010), 1493-1547, arXiv:0803.2430.
  8. Andruskiewitsch N., Peña Pollastri H.M., On the double of the (restricted) super Jordan plane, New York J. Math. 28 (2022), 1596-1622, arXiv:2008.01234.
  9. Andruskiewitsch N., Sanmarco G., Finite GK-dimensional pre-Nichols algebras of quantum linear spaces and of Cartan type, Trans. Amer. Math. Soc. Ser. B 8 (2021), 296-329, arXiv:2002.11087.
  10. Andruskiewitsch N., Schneider H.J., Pointed Hopf algebras, in New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., Vol. 43, Cambridge University Press, Cambridge, 2002, 1-68, arXiv:math.QA/0110136.
  11. Angiono I., Campagnolo E., Sanmarco G., Finite GK-dimensional pre-Nichols algebras of super and standard type, arXiv:2009.04863.
  12. Angiono I., García Iglesias A., On finite GK-dimensional Nichols algebras of diagonal type: rank 3 and Cartan type, Publ. Mat., to appear, arXiv:2106.10143.
  13. Angiono I., García Iglesias A., Finite GK-dimensional Nichols algebras of diagonal type and finite root systems, arXiv:2212.08169.
  14. Brown K.A., Zhang J.J., Survey on Hopf algebras of GK-dimension 1 and 2, in Hopf Algebras, Tensor Categories and Related Topics, Contemp. Math., Vol. 771, Amer. Math. Soc., Providence, RI, 2021, 43-62, arXiv:2003.14251.
  15. Graña M., A freeness theorem for Nichols algebras, J. Algebra 231 (2000), 235-257.
  16. Heckenberger I., Classification of arithmetic root systems, Adv. Math. 220 (2009), 59-124, arXiv:math.QA/0605795.
  17. Heckenberger I., Schneider H.J., Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras, Adv. Math. 244 (2013), 354-394, arXiv:1111.4673.
  18. Heckenberger I., Schneider H.J., Hopf algebras and root systems, Math. Surveys Monogr., Vol. 247, Amer. Math. Soc., Providence, RI, 2020.
  19. Krause G.R., Lenagan T.H., Growth of algebras and Gelfand-Kirillov dimension, Grad. Stud. Math., Vol. 22, Amer. Math. Soc., Providence, RI, 2000.
  20. Rosso M., Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399-416.
  21. Takeuchi M., Survey of braided Hopf algebras, in New Trends in Hopf Algebra Theory (La Falda, 1999), Contemp. Math., Vol. 267, Amer. Math. Soc., Providence, RI, 2000, 301-323.
  22. Ufer S., PBW bases for a class of braided Hopf algebras, J. Algebra 280 (2004), 84-119, arXiv:math.QA/0311504.

Previous article  Next article  Contents of Volume 19 (2023)